Scalefree hardness of average-case Euclidean TSP approximation

Approximation Hardness of Graphic TSP on Cubic Graphs

We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used in the paper could be also of independent interest.

New Inapproximability Bounds for TSP

In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to Papadimitriou and Vempala). We construct here two new bounded occurrence CSP reductions which improve these bounds to 123/122 and 75/74, respectively. The latter bound is the first improvement in more than a decade for the case of the asymmetric TSP. One of our main tools, which may be of…

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $α\ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^α$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,α$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{α-1}+\sqrt{6}^α/3$ for $d=2$ and all $α\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomi…

增强的组合结构生成：近似难度

Reinforced Generation of Combinatorial Structures: Hardness of Approximation

利用AI方法在复杂性理论中取得进展，改进MAX-CUT、MAX-3-CUT和TSP的近似难度结果。

条件复杂性难度：单调电路大小、矩阵刚性与张量秩

Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank

本文基于条件复杂性理论，通过非确定性下界构造高复杂度的布尔函数、矩阵和张量，证明了电路大小的下界。